Understanding Derivatives of f(r) in Multiple Variables

  • Thread starter Jhenrique
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In summary, the conversation discusses the computation of derivatives with respect to independent variables. The chain rule is used when there are multiple variables, and when the variables are nested, the chain rule is repeated. There may be a more compact way to differentiate using vectors, such as ##\frac{d^2f}{du dv}=\frac{d^2 f}{d\vec{r}^T d\vec{r}}\cdot \frac{d\vec{r}}{du}\times \frac{d\vec{r}}{dv}##. However, this method is not fully understood and the speaker suggests using partial derivatives when there are multiple variables.
  • #1
Jhenrique
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I have some important questions and essentials for understand some theories. They are six:
given f(r(t)), f(r(u, v)), f(r(u(t), v(t))), f(r(t)), f(r(u, v)) and f(r(u(t), v(t))). How compute its derivatives wrt independent variables?

Unfortunately, I just know the answer for 1nd:
[tex]\frac{df}{dt}=\frac{df}{d\vec{r}}\cdot \frac{d\vec{r}}{dt}[/tex]
I don't know equate the other derivatives.
 
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  • #2
Where you have more than one variable, you use partial derivatives.
When the variables are nested, just repeat the chain rule.
Recall how to differentiate a vector.
 
  • #3
But I think that chain rule will expand to much. I think if exist another way more compact for these differentations... some so so like
##\frac{d^2f}{du dv}=\frac{d^2 f}{d\vec{r}^T d\vec{r}}\cdot \frac{d\vec{r}}{du}\times \frac{d\vec{r}}{dv}##
 

FAQ: Understanding Derivatives of f(r) in Multiple Variables

What is the purpose of understanding derivatives of f(r) in multiple variables?

The purpose of understanding derivatives of f(r) in multiple variables is to analyze and quantify the rate of change of a function with respect to multiple variables. This is important in various fields such as physics, economics, and engineering, as it allows for more accurate predictions and optimizations.

How are derivatives of f(r) in multiple variables calculated?

Derivatives of f(r) in multiple variables are calculated using the partial derivative concept, where each variable is treated as a constant while taking the derivative with respect to the variable of interest. This results in a partial derivative for each variable, which can then be combined to find the overall derivative.

What is the difference between a partial derivative and an ordinary derivative?

A partial derivative calculates the rate of change of a function with respect to a specific variable, while holding all other variables constant. On the other hand, an ordinary derivative calculates the rate of change of a function with respect to a single variable. In other words, partial derivatives take into account multiple variables, while ordinary derivatives focus on one variable.

How can understanding derivatives of f(r) in multiple variables be applied in real-world situations?

Understanding derivatives of f(r) in multiple variables can be applied in various real-world situations, such as analyzing the cost and revenue functions in economics, optimizing production processes in engineering, and predicting the motion of objects in physics. It is a powerful tool for making accurate predictions and optimizing systems.

Are there any limitations to using derivatives of f(r) in multiple variables?

One limitation of using derivatives of f(r) in multiple variables is that it assumes a continuous and smooth relationship between variables. This may not always be the case in real-world situations, leading to inaccurate predictions. Additionally, calculating derivatives can become more complex as the number of variables increases, making it difficult to apply in certain situations.

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